As there are many ways to try to explain
these twisty pieces of logic, we thought that we'd give the correct
explanations to each part of the problem in turn. Which of the
solutions do you like best?

This problem was also discussed on the
askNRICH pages. You might like to read about this discussionhere .

Especially well done to those who made it
through to the end of the question and (hopefully) a well earned
mental rest from these questions which seem to be both easy and
hard at the same time ...

The surprise Test

Ashley said: There is a
logic you could think of that could mean you can't have it on any
day. You can think it can't be on Friday because, if it wasn't done
until then, it wouldn't be a surprise on the Friday. But, if it
definitely isn't on Friday, you could argue that it won't be a
surprise if you have it on Thursday as it wouldn't have been on the
other 3 days and it can't be on Friday. What's more, the same rule
could apply to all other days, meaning it's impossible for every
day of the week.

Jenny said: The test cannot
be given on the Friday, as she said that they would only know on
the morning of the test. But this means that it also cannot be
given on the Thursday either for the same reason.

Hannah wrote: If the test
still has not been given on Thursday, that means they will know the
test will be on a Friday. So Friday is out. So if they still
haven't been given the test by Wednesday, it means it will be on
Thursday. So the test cannot be on Thursday. If the test hasn't
been given on Tuesday, they will know it is on Wednesday. If the
test hasn't been given on Monday, then it has to be on Tuesday. So
the test has to be on Monday, but the class will know that too. So
the test can't be given.

B Cole from Gt Ellingham Primary
wrote in to say: Loved this puzzle, thankyou. (pleased to hear it! ) Eventually realised that
the test couldn't be given on the Friday (as pupils would know at
the end of Thursday), and that, by the same logic, the test can not
be given on Thursday.

We really liked
the way that B Cole's solution continued in the 'real' world with
'real children':
But my TA pointed out that most of the pupils could be given the
test on the Thursday, because only the brightest ones would have
worked out the above. And, a certain number of kids could be given
the test on the Friday, as they would have forgotten all about it!

Great work from the Teaching Assistant,
whoever you are.

Laura said: The test cannot
be given on Friday, since the students would all know by the end of
Thursday. So the test can only be given Monday to Thursday. However
by the same logic, the test cannot possibly be on Thursday, since
all the students would know by the end of Wednesday. So Thursday is
ruled out. Using this logic, Wednesday is ruled out too, as is
Tuesday, at which point the test could not possibly be next week,
as the students would know that the test would be on Monday.

Qiuying said: If he test
will be given on Friday, then the fact that the teacher said "they
will not know which day until they are told on the morning of the
test" will be untrue. If the test wasn't given by Wednesday, then
since the test wouldn't be given on Friday, then the children will
know that it must be given on Thursday. It the test wasn't given by
Tuesday, then the children will know that it is going to be on
Wednesday and so on. Therefore, according to what the teacher said,
the children can always predict that the test will be given on
Monday and the teacher was contradicting herself because, if she
was telling the truth, then none of the children will know which
day the test was on, but at the same time she was giving away the
day which the test will be taken which makes her saying untrue.

Michelle also noticed that the
test couldn't be on any day and added: So the students who
figured this out will be relaxing, knowing that the test can't be
this week. But, the cunning teacher (who probably already know
this) will set the test at Friday or some other day. So, overall,
the teacher's statemant becomes true after all. We hadn't thought about this
veryinteresting point: If the students reason that the test cannot
happen, then they will certainly be surprised if it happens on any
day at all. Excellent work, Michelle!

The universal reference book

Simeon said both Yes and No

Patrick noted UltraRef is a
paradox - if it does not refer to itself then it must
because it refers to all books that do not refer to themselves; if
it does refer to itself then it should not be included.

Patrick
suggested a resolution to this paradox: However, this can be
solved by assuming that UltraRef can refer to books that refer to
themselves, as well.

Jenny said: The reference
book cannot have itself in its index unless it does not have itself
in the index. Then when it has itself in the index it cannot have
itself in the index, so in the end it cannot have itself in the
index, but then it must have itself in its index....

Hannah said: Does UltraRef
refer to itself in its index? Well, if it doesn't, then it will
have to add itself to its index, because otherwise it will not
refer to every single book. However, once it is added to the index,
it will refer to itself in its index, so it will have to be removed
from the index. But then, it will have to be added again and so on.

Laura very neatly said:
UltraRef cannot refer to itself in the index, since the index lists
only those books which do not reference themselves in the index.
However if Ultra Ref does not reference itself in the index, then
it SHOULD be listed in its index, for not listing itself, which is
a contradiction in itself.

Smallest number paradox

Patrick said : This cannot
be fulfilled - the definition of N is The smallest whole number not
definable in under eleven words, but this is ten words and
describes N perfectly.

Laura said : N cannot be
defined as ''The smallest whole number not definable in under
eleven words'', since this definition in only nine words long, and
the number N has just been defined.

Hannah shows that you can
interpret 'smallest' in a different way : This is
impossible. Negative numbers are whole numbers as well.

In all of these questions we need to assess
closely the linguistic meaning of the words in the sentence. In
mathematics, we always try to make things as clear as possible, but
ambiguity (more than one possible meaning) can still arise.

The liar

Patrick said : This is a
paradox - if he is lying then he is saying he can only ever speak
the truth, but he has lied already. If he is telling the truth then
he is saying he always lies, but he has just spoken the truth. This
can be solved by taking the opposite of "I always lie" to mean "I
don't always lie" and interpreting this as "I sometimes lie".

Hannah said : That is
impossible. If he only lies, this means the sentence is true, which
means he only lies, which means the sentence is true, which
means....

Laura very clearly said :
The person saying "I only ever lie" cannot be telling the truth, as
he is admitting to never being truthful. Neither can he be lying,
because then the interpretation of the phrase under these
circumstances is "I only ever tell the truth", which is not so,
since he is lying.

Qiuying said : If the
friend is telling the 'truth', then what he said must be true, so
he should 'only ever lie' which means that he is a liar that never
tells the truth. So this statement obviously contradicts itself.
However, if the friend was telling a lie, then the fact that he
'only ever lies' remains true which also contradicts itself.

Note that the sentence 'I only ever lie'
cannot be true. If it is not true then we need to be clear what
this means. Does it mean 'I only ever tell the truth', or does it
mean 'I sometimes tell the truth'? One of these interpretations
leads to paradox, one does not.

The false sentence

Hannahnoticed that this question was essentially the same as
the previous question -- great observation.

Patrick wrote : This
sentence, if false, means that it is true, which it is not. If it
is true, then it says it is false, which it is not.

Laura reasoned: If it is
true that the sentence is false, then the statement is true, and if
it is incorrect that the sentence is true, then the statement is
false. It is a contradiction in terms.

Quiyang said : If the
sentence was 'true', then at the same time it is saying:'this
sentence is false' and is contradicting itself. However, if the
sentence was 'false', then it's saying--'this sentence was false'
is true, which is also against eachother.

The erroneous statement:

Patrick thought : There are
two obvious errors (the double letters), but the third is the
meaning: the sentence assumes that the third error is the fact that
there are only two errors - but this makes the "three errors" part
wrong, as the third error is not an error.

Hannah correctly saw : The
first error is the word "three". The second is the word "errors".
The third error is the whole statement. There are only two errors.
Which makes it into three errors again. And then two. And then
three

Laura clearly said : The
three errors are 2 typing mistakes and seemingly no others.
However, the fact that the third error - that there is no error -
suggests that the statement is indeed true. However by the
statement being true, there are now only the 2 spelling errors and
no error in the meaning.